57 research outputs found
Distributed solution of Laplacian eigenvalue problems
The purpose of this article is to approximately compute the eigenvalues of
the symmetric Dirichlet Laplacian within an interval . A novel
domain decomposition Ritz method, partition of unity condensed pole
interpolation method, is proposed. This method can be used in distributed
computing environments where communication is expensive, e.g., in clusters
running on cloud computing services or networked workstations. The Ritz space
is obtained from local subspaces consistent with a decomposition of the domain
into subdomains. These local subspaces are constructed independently of each
other, using data only related to the corresponding subdomain. Relative
eigenvalue error is analysed. Numerical examples on a cluster of workstations
validate the error analysis and the performance of the method.Comment: 28 page
Efficient solution of symmetric eigenvalue problems from families of coupled systems
Efficient solution of the lowest eigenmodes is studied for a family of
related eigenvalue problems with common block structure. It is
assumed that the upper diagonal block varies between different versions while
the lower diagonal block and the range of the coupling blocks remains
unchanged. Such block structure naturally arises when studying the effect of a
subsystem to the eigenmodes of the full system. The proposed method is based on
interpolation of the resolvent function after some of its singularities have
been removed by a spectral projection. Singular value decomposition can be used
to further reduce the dimension of the computational problem. Error analysis of
the method indicates exponential convergence with respect to the number of
interpolation points. Theoretical results are illustrated by two numerical
examples related to finite element discretisation of the Laplace operator
Edge-promoting reconstruction of absorption and diffusivity in optical tomography
In optical tomography a physical body is illuminated with near-infrared light
and the resulting outward photon flux is measured at the object boundary. The
goal is to reconstruct internal optical properties of the body, such as
absorption and diffusivity. In this work, it is assumed that the imaged object
is composed of an approximately homogeneous background with clearly
distinguishable embedded inhomogeneities. An algorithm for finding the maximum
a posteriori estimate for the absorption and diffusion coefficients is
introduced assuming an edge-preferring prior and an additive Gaussian
measurement noise model. The method is based on iteratively combining a lagged
diffusivity step and a linearization of the measurement model of diffuse
optical tomography with priorconditioned LSQR. The performance of the
reconstruction technique is tested via three-dimensional numerical experiments
with simulated measurement data.Comment: 18 pages, 6 figure
Finite element methods for time-harmonic wave equations
This thesis concerns the numerical simulation of time-harmonic wave equations using the finite element method. The main difficulties in solving wave equations are the large number of unknowns and the solution of the resulting linear system. The focus of the research is in preconditioned iterative methods for solving the linear system and in the validation of the result with a posteriori error estimation. Two different solution strategies for solving the Helmholtz equation, a domain decomposition method and a preconditioned GMRES method are studied. In addition, an a posterior error estimate for the Maxwell's equations is presented.
The presented domain decomposition method is based on the hybridized mixed Helmholtz equation and using a high-order, tensorial eigenbasis. The efficiency of this method is demonstrated by numerical examples. As the first step towards the mathematical analysis of the domain decomposition method, preconditioners for mixed systems are studied. This leads to a new preconditioner for the mixed Poisson problem, which allows any preconditioned for the first order finite element discretization of the Poisson problem to be used with iterative methods for the Schur complement problem.
Solving the linear systems arising from the first order finite element discretization of the Helmholtz equation using the GMRES method with a Laplace, an inexact Laplace, or a two-level preconditioner is discussed. The convergence properties of the preconditioned GMRES method are analyzed by using a convergence criterion based on the field of values. A functional type a posterior error estimate is derived for simplifications of the Maxwell's equations. This estimate gives computable, guaranteed upper bounds for the discretization error
Inverse heat source problem and experimental design for determining iron loss distribution
Iron loss determination in the magnetic core of an electrical machine, such
as a motor or a transformer, is formulated as an inverse heat source problem.
The sensor positions inside the object are optimized in order to minimize the
uncertainty in the reconstruction in the sense of the A-optimality of Bayesian
experimental design. This paper focuses on the problem formulation and an
efficient numerical solution of the discretized sensor optimization and source
reconstruction problems. A semirealistic linear model is discretized by finite
elements and studied numerically.Comment: 30 pages, 15 figure
Computational framework for applying electrical impedance tomography to head imaging
This work introduces a computational framework for applying absolute
electrical impedance tomography to head imaging without accurate information on
the head shape or the electrode positions. A library of fifty heads is employed
to build a principal component model for the typical variations in the shape of
the human head, which leads to a relatively accurate parametrization for head
shapes with only a few free parameters. The estimation of these shape
parameters and the electrode positions is incorporated in a regularized
Newton-type output least squares reconstruction algorithm. The presented
numerical experiments demonstrate that strong enough variations in the internal
conductivity of a human head can be detected by absolute electrical impedance
tomography even if the geometric information on the measurement configuration
is incomplete to an extent that is to be expected in practice.Comment: 25 pages, 12 figure
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